PomeA10584C44FB2The radial energy is built from the Monge–Ampère one-form, and its first variation exhibits the torsion of the background. When a Hermitian form is the $(1,1)$-part of a closed real two-form, there is another natural energy: the squared $L^2$ norm of the smallest $(2,0)$-part needed to close it. In complex dimension four, adding a complex Hessian to the form changes this energy by minus the change of the volume. Its potential variations are therefore governed by the volume, and not by a primitive of the Monge–Ampère one-form.
Throughout this section we assume $n=4$. A Hermitian form $\omega $ is Hermitian symplectic if there is a smooth $(2,0)$-form $\rho $, called a completion of $\omega $, such that
Equivalently, the real two-form $\omega +\rho +\bar\rho $ is closed. Applying $\bar\partial $ to the second equation shows that $\omega $ is pluriclosed, meaning $\mathrm{dd}^{\mathrm{c}}\omega =0$. Two completions differ by a closed holomorphic $(2,0)$-form, so the completions of $\omega $ form an affine space over the finite-dimensional space of closed holomorphic $(2,0)$-forms. The quadratic norm
is positive definite on $(2,0)$-forms, and minimizing it over this affine space gives a unique canonical completion $\rho _\omega $. It is characterized by being orthogonal to every closed holomorphic $(2,0)$-form. Define the completion energy and the volume of $\omega $ by
The completion energy is nonnegative and vanishes exactly when $\omega $ is Kähler: a zero canonical completion forces $\partial \omega =0$, while a closed $\omega $ has the zero completion.
Since $\omega $ is pluriclosed, it defines an Aeppli cohomology class, whose real $(1,1)$-representatives are the forms $\omega +\partial \bar\tau +\bar\partial \tau $, with $\tau $ a smooth $(1,0)$-form. We call a real form $\psi =\partial \bar\tau +\bar\partial \tau $ an Aeppli variation. If $\rho $ completes $\omega $ and $\omega +t\psi $ is positive for a real $t$, then $\omega +t\psi $ has the completion $\rho +t\partial \tau $. In particular, the positive representatives $\omega +\partial \bar\tau +\bar\partial \tau >0$ of the Aeppli class of $\omega $ are again Hermitian symplectic. Every positive potential variation $\omega +\mathrm{dd}^{\mathrm{c}}\varphi $ belongs to this class and has the same exterior derivative as $\omega $. We write $DF_\omega (\psi )$ for the derivative of $t\mapsto F(\omega +t\psi )$ at $t=0$.
Lemma 11.2.15.19D4882CC9D5 Let $\omega $ be Hermitian symplectic, with canonical completion $\rho =\rho _\omega $. Every Aeppli variation $\psi =\partial \bar\tau +\bar\partial \tau $ satisfies
7C73F0133C9D
11.2.16
Proof. The canonical completion of $\omega +t\psi $ is $\rho +t\partial \tau -h(t)$, where $h(t)$ is a closed holomorphic $(2,0)$-form and $h(0)=0$. In a fixed basis of these holomorphic forms, the orthogonal projection defining $h(t)$ solves a positive definite Gram system whose coefficients are smooth in $t$. Since $\rho $ is orthogonal to closed holomorphic $(2,0)$-forms, the contribution of $h'(0)$ to the first derivative of the squared norm vanishes. We may therefore differentiate using the completion $\rho +t\partial \tau $. Differentiating its two factors and the factor $(\omega +t\psi )^2/2$ gives the formula. $\square$
Proposition 11.2.17.AE6CE389C569 Let $\omega $ be Hermitian symplectic. For every smooth real function $\varphi $ with $\omega _\varphi =\omega +\mathrm{dd}^{\mathrm{c}}\varphi >0$, one has $\rho _{\omega _\varphi }=\rho _\omega $ and
BAFFEA611460
11.2.18
Moreover, $DF_\omega (\mathrm{dd}^{\mathrm{c}}\varphi )=0$ for every smooth real $\varphi $ if and only if $\mathrm{dd}^{\mathrm{c}}(\omega ^3)=0$. Thus a critical point of $F$ in its Aeppli class is Gauduchon.
Proof. Fix any completion $\zeta $ of $\omega $ and put $\Theta _\zeta =\omega +\zeta +\bar\zeta $. Expansion by type gives
Since $\partial \omega _\varphi =\partial \omega $, the forms $\omega _\varphi $ and $\omega $ have the same completions. With $\zeta $ fixed, replacing $\omega $ by $\omega _\varphi $ changes the closed form $\Theta _\zeta $ by an exact form, so the integral on the left is unchanged; so is the last integral on the right, which does not involve $\omega $. Hence
This difference is independent of $\zeta $, so the two norms have the same minimizer, which gives $\rho _{\omega _\varphi }=\rho _\omega $ and (11.2.18). Differentiating at $t=0$ along $\omega +t\mathrm{dd}^{\mathrm{c}}\varphi $ and using lemma 11.1.2, we obtain
Since $\varphi $ is arbitrary, this proves the remaining assertions. $\square$
The positive index of a real quadratic form on a real vector space is the supremum of the dimensions of the finite-dimensional subspaces on which it is positive definite; the negative index is defined in the same way. Thus the positive index is infinite when the form is positive definite on subspaces of arbitrarily large finite dimension.
Theorem 11.2.19.04BD04D6C676 Let $\omega $ be Hermitian symplectic, and let $H_\omega $ be the Hessian of $F$ at $\omega $ along the affine space of real representatives of the Aeppli class of $\omega $, so that $H_\omega (\psi ,\psi )$ is the second derivative of $t\mapsto F(\omega +t\psi )$ at $t=0$. If $\mathrm{dd}^{\mathrm{c}}(\omega ^2)\ne 0$, the restriction of $H_\omega $ to potential directions $\psi =\mathrm{dd}^{\mathrm{c}}\varphi $ has infinite positive and negative index. If $\mathrm{dd}^{\mathrm{c}}(\omega ^2)=0$, then $F(\omega +\mathrm{dd}^{\mathrm{c}}\varphi )=F(\omega )$ whenever $\omega +\mathrm{dd}^{\mathrm{c}}\varphi >0$. Consequently, every local minimum of $F$ in an Aeppli class satisfies $\mathrm{dd}^{\mathrm{c}}(\omega ^2)=0$.
Proof. Step 1: the Hessian formula. Pluriclosedness and Stokes' theorem give $\int _ X\omega \wedge (\mathrm{dd}^{\mathrm{c}}\varphi )^3=\int _ X(\mathrm{dd}^{\mathrm{c}}\varphi )^4=0$. Thus the volume and, by (11.2.18), the energy along a potential line $t\mapsto \omega +t\mathrm{dd}^{\mathrm{c}}\varphi $ are polynomials of degree at most two in $t$. Differentiating twice and integrating by parts, we obtain
8B6CFC8EA37B
11.2.20
For the second equality, apply lemma 11.1.2 once and then use the identity $\mathrm{dd}^{\mathrm{c}}(\varphi ^2/2)=\varphi \mathrm{dd}^{\mathrm{c}}\varphi +\mathrm d\varphi \wedge \mathrm d^{\mathrm c}\varphi $ against the closed form $\mathrm{dd}^{\mathrm{c}}(\omega ^2)$.
Step 2: both indices are infinite if $\mathrm{dd}^{\mathrm{c}}(\omega ^2)\ne 0$. Put $\Psi =\mathrm{dd}^{\mathrm{c}}(\omega ^2)$. At each point, $\xi \mapsto \Psi \wedge \mathrm i\xi \wedge \bar\xi $, divided by a positive volume form, is a Hermitian quadratic form in the $(1,0)$-covector $\xi $, and these quadratic forms determine $\Psi $. Their traces with respect to $\omega $ are positive multiples of the density $\Psi \wedge \omega $, whose integral vanishes by Stokes' theorem and $\mathrm{dd}^{\mathrm{c}}\omega =0$. If the quadratic forms took no strictly positive value, they would be negative semidefinite everywhere, with nonpositive traces of integral zero; the traces, and hence $\Psi $, would vanish identically. The same argument applies with the signs reversed. Therefore, if $\Psi \ne 0$, the quadratic forms take a strictly positive value somewhere and a strictly negative value somewhere.
Fix one of the two signs. Choose holomorphic coordinates on a ball such that, with $x=\operatorname {Re}z_1$, the density $\Psi \wedge \mathrm dx\wedge \mathrm d^{\mathrm c}x$ has that sign on the ball; this is possible because $\mathrm dx\wedge \mathrm d^{\mathrm c}x$ is a positive multiple of $\mathrm i\, \mathrm dz_1\wedge \mathrm d\bar z_1$. For a nonzero real cutoff function $a$ supported in the ball, set $\varphi _ N=a\cos (Nx)$. Formula (11.2.20) gives
where the constant in $O(N)$ depends on $\Psi $, the chart and the cutoff, but not on $N$. By integration by parts in $x$, the integral tends to half the same integral without the factor $\sin ^2(Nx)$. Hence the Hessian has the chosen sign for large $N$. For a positive integer $k$, apply this construction on $k$ disjoint smaller balls in that region. By the local integral formula, the resulting nonzero directions are linearly independent and pairwise orthogonal for the Hessian, so their span is a $k$-dimensional subspace on which $H_\omega $ is definite of the chosen sign. Repeating the argument with the opposite sign proves that both indices are infinite. Each finite-dimensional span admits sufficiently small variations preserving positivity.
Step 3: flatness if $\mathrm{dd}^{\mathrm{c}}(\omega ^2)=0$. If $\Psi =0$, lemma 11.1.3 and pluriclosedness give $\mathrm{dd}^{\mathrm{c}}(\omega ^3)=3\omega \wedge \Psi =0$. Both derivatives at $t=0$ of the quadratic polynomial $t\mapsto F(\omega +t\mathrm{dd}^{\mathrm{c}}\varphi )$ vanish, so it is constant on the interval of $t$ for which $\omega +t\mathrm{dd}^{\mathrm{c}}\varphi >0$. This interval contains $0$ and $1$ whenever $\omega +\mathrm{dd}^{\mathrm{c}}\varphi >0$. Finally, a local minimum cannot have a potential direction along which the Hessian is negative, so it satisfies $\mathrm{dd}^{\mathrm{c}}(\omega ^2)=0$ by Step 2. $\square$
Corollary 11.2.21.67DF1C6C0509 For a Hermitian symplectic form $\omega $ on $X$, the following conditions are equivalent:
$F(\omega +\mathrm{dd}^{\mathrm{c}}\varphi )$ is independent of the smooth real function $\varphi $ with $\omega +\mathrm{dd}^{\mathrm{c}}\varphi >0$;
$\mathrm{dd}^{\mathrm{c}}(\omega ^2)=0$;
$\int _ X(\omega +\mathrm{dd}^{\mathrm{c}}u)^4$ is independent of the bounded $\omega $-plurisubharmonic function $u$.
The common mass in the third condition is $24V(\omega )$. Every local minimum of $F$ in an Aeppli class has these properties.
Proof. By (11.2.18), the energy is constant along smooth positive potential variations if and only if their volume is. Since $\mathrm{dd}^{\mathrm{c}}\omega =0$, lemma 11.1.3, theorem 11.1.7 show that this smooth mass invariance is equivalent to $\mathrm{dd}^{\mathrm{c}}(\omega ^2)=0$ and to mass invariance for bounded potentials. The zero potential gives the common mass. The last assertion follows from theorem 11.2.19. $\square$
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